Plastic collapse in non-associated hardening materials with application to Cam-clay

Abstract

This paper presents a variational formulation for the analysis of plastic collapse conditions for a class of hardening materials that accounts for some non-associated flow laws such as the modified Cam-clay model of soils. In this framework, classical statical and kinematical principles of limit analysis do not hold. The variational principle is formulated for the general class of materials whose flow equations are derived from a kind of generalized potentials named bipotentials by de Saxcé.The plastic collapse phenomenon for hardening materials is considered first and formulated as a system of equations. In particular, the case of the usual modified Cam-clay model is analyzed. The paper follows with the proposal of a minimization principle whose solution is then related to the solution of the plastic collapse problem. We demonstrate the use of this minimum principle in a simple example of triaxial compression of a modified Cam-clay material. Finally, we discuss the particular form of the proposed variational formulation for the case of associated plasticity.

A variational approach via bipotentials for unilateral contact problems

“…Due to (5), u ∈ U 0 . Therefore, using again (15) and (18) we obtain b(u, µ) ≥ (f , u) V for all µ ∈ Λ.…”

“…The key of the proposed approach is a bipotential function which depends on the constitutive map and its Fenchel conjugate. The construction of several bipotential functions appears in connection with Coulomb's friction law [3] and Cam-Clay models in soil mechanics [4,5], cyclic plasticity [6,7] and viscoplasticity of metals with nonlinear kinematical hardening rule [8],…”

On the superlinear convergence in computational elasto-plasticity

“…Modified Cam-clay is an enhancement of the original Cam-clay model, and, in contrast to its predecessor, has a smooth yield surface, being an ellipse in the deviatoric-hydrostatic plane. We mainly follow the presentation given in Borja and Lee (1990) and Zouain et al (2007). In the former, also a return algorithm in order to compute the response function and the consistent tangent is given.…”

A variational principle of elastoplasticity and its application to the modeling of frictional materials

“…Indeed, even the basic Cam clay models (which would require S ¼ 0) do not permit a variational formulation since the hardening law is nonassociated (see e.g. Ortiz and Pandolfi, 2004;Hjiaj et al, 2005;Zouain et al, 2007). In the following, therefore, we consider an incremental variational formulation.…”